This article presents a generalization of previous theorems on common fixed points of self-maps. It introduces the concept of property E.A. and weak compatibility between self-maps. A new theorem (Theorem B) is proved which finds a unique common fixed point for three self-maps under weaker conditions than previous results, including relaxing orbital completeness and removing the requirement of orbital continuity. The proof of Theorem B is provided. It is shown that this new theorem generalizes an earlier result from the literature.

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Vol.3, No.8, 2013
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A Weaker Version of Continuity and a Common Fixed Point
Theorem
Swatmaram
Chaitanya Bharathi Institute of Technology, Hyderabad-500075, Andhra Pradesh State, India
e-mail: ramuswatma@yahoo.com,
T. Phaneendra
Applied Analysis Division, School of Advanced Sciences, VIT University, Vellore-632014, Tamil Nadu State,
India, e-mail: drtp.indra@gmail.com
Abstract
A generalization of results of the second author [4] and the authors [5] have been proved through the notions of
property E.A. and weak compatibility and restricting the orbital completeness of the space.
2000 AMS mathematics subject classification: 54 H 25
Key Words: Compatible self-maps, weakly compatible self-maps, property EA and common fixed point
1. Introduction
Let (X, d) be a metric space. Given Xx0 ∈ and self-maps A, S and T on X, if there exist points x1,
x2, …, xn, … in X such that
,AxSx 1n22n2 −− = n21n2 AxTx =− for n = 1, 2, 3, …, … (1)
Then the sequence
∞
=1nnAx is an ( )T,S -orbit at x0 with respect to A.
Definition 1.1. The space X is ( )T,S - orbitally complete with respect to A at x0 [4] if every Cauchy sequence in
some orbit of the form from equation (1) converges in X.
Definition 1.2. The pair ( )TS, is Asymptotically regular at x0 with respect to A [4] if the orbit (1) satisfies the
condition that ( ) 0AxAxd 1nn →+ + as ∞→n .
Definition 1.3. Self-map A on X is orbitally continuous at x0 if it is continuous at every point of some orbit at x0.
Obviously every continuous self-map on X is orbitally continuous at each Xx0 ∈ . However the
converse is not true as seen from [4].
Definition 1.4. Self-maps A and S are compatible [2] if ( ) 0SAx,ASxdlim nnn
=∞→
whenever
( ) 0Ax,Sxdlim nnn
=→∞
.
Definition 1.5. Self-maps A and S are said to be weakly compatible [3] if they commute at their coincidence
points.
With these notions the following theorem was proved in [4]:
Theorem A: Let A, S and T be self-maps on X satisfying the inequality.
( ) ( ) ( ) ( ) ( ) ( ){ }Sx,Ayd,Ty,Axd,Ty,Ayd,Sx,Axd,Ay,AxdmaxcTy,Sxd ≤
for all Xy,x ∈ ,… (2)
where .1c0 <≤ Suppose that at Xx0 ∈ ,
(a) The pair ( )T,S is asymptotically regular with respect to A.
(b) The space X is orbitally complete
(c) A is orbitally continuous
(d) Either ( )S,A or ( )T,A is compatible pair.
Then A, S and T will have a unique common fixed point.
In this paper, we prove a generalization of Theorem A by using the property EA (cf. Section
2), relaxing the condition (b), removing the condition (c) and weakening the condition (d).
2. Main result
Definition 2.1. Self-maps A and S satisfy property E.A. [1] if there exists a sequence Xx 1nn ⊂
∞
=
such that

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Vol.3, No.8, 2013
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zSxlimAxlim nnnn
== ∞→∞→
.
Theorem B: Let A, S and T be self-maps on X satisfying the inequality (2).
Suppose that
(e) either ( )S,A or ( )T,A satisfies property E.A.
(f) A(X) is orbitally complete subspace of X.
and
(g) ( )S,A or ( )T,A is weakly compatible.
Then all the three self-maps will have a unique common fixed point.
Proof. By the property EA for the pair ( )S,A we have
zSxlimAxlim nnnn
== ∞→∞→
for some .Xz∈ … (3)
Let pTxlim nn
=∞→
. Now we prove that p = z.
In fact from the inequality (2) with nxx = and nxy = we get
( ) ( ) ( ) ( ) ( ){ }nnnnnnnnnnnn Sx,Axd,Tx,Axd,Tx,Axd,Sx,Axd,Ax,Axdmaxc)Tx,d(Sx ≤
applying the limit as ∞→n and then using (3), this gives
( ) ( ) ( ){ }0,p,zd,p,zd,0,0maxcp,zd ≤ or ( ) ( )p,zdcp,zd ≤ so that p = z.
Thus .zTxlimSxlimAxlim nnnnnn
=== ∞→∞→∞→
… (4)
Equation (4) can also be obtained in similar lines whenever ( )T,A satisfies property E.A.
From the orbital completeness (f) we see that ( )XAz∈ so that Auz = for some .Xu ∈ .
Now, taking ux = and nxy = in (2),
( ) ( ) ( ) ( ) ( ){ }Su,Axd,Tx,Aud,Tx,Axd,Su,Aud,Ax,Audmaxc)Txd(Su, nnnnnn ≤
Applying the limit as ,n ∞→ using (4) and zAu = in this, we get
( ) ( ) ( ){ }Su,Aud0,0,Su,Aud,0maxcAu,Sud ≤ ⇒ ( ) ( )Su,AudcAu,Sud ≤ or .zSuAu ==
Then from the weak compatibility of )S,A( , we see that SAuASu = or SzAz = .
Again writing zyx == in the inequality (2) and using Az = Sz, it follows that
( ) ( ) ( ){ } ( )Tz,Szcd0,Tz,Szd,Tz,Szd,0,0maxcTz,Szd =≤ so that TzSz = .
That is, TzSzAz == . … (5)
Taking zy,xx n == in (2), we have
( ) ( ) ( ) ( ) ( ){ }nnnnnn Sx,Azd,Tz,Axd,Tz,Azd,Sx,Axd,Az,AxdmaxcTz),d(Sx ≤
As limit ∞→n , this along with (4) and (5) implies that
( ) ( ) ( ) ( ) ( ){ } ( )Tz,zcdz,Tzd,Tz,zd,Az,Azd,z,zd,Tz,zdmaxcTz)d(z, =≤ or z = Tz.
Thus z is a common fixed point of self-maps A, S and T.
On the other hand, with minor changes in the above proof we can prove that Au = Tu = z.
Suppose that the pair ( )T,A is weakly compatible. Then it follows that TAuATu = or Az = Tz.
Proceeding as in the previous steps, we get that zSzTzAz === .
Uniqueness: Let z, z′ be two common fixed points of A, S, and T. Then from (2) with x = z and y = z′, we get
( ) ( ) ( ) ( ) ( ) ( ){ }Sz,'Azd,'Tz,Azd,'Tz,'Azd,Sz,Azd,'Az,Azdmaxc'Tz,Szd ≤
⇒ ( ) ( ) ( ) ( ) ( ) ( ){ } ( )'z,zcdz,'zd,'z,zd,'z,'zd,z,zd,'z,zdmaxc'z,zd =≤ or z = z′.
Hence the fixed point is unique.
Remark 2.1. Suppose at some Xx0 ∈ , (a) holds good then from the proof of Theorem A, the sequnce
∞
=1nnAx defined by (1) is Cauchy and hence by the orbital completeness, we can find some Xw ∈ such that
,wTxlimAxlimSxlimAxlim 1n2n2n2nn2nn2n
==== +∞→+∞→∞→∞→
which immediately implies that the pairs )T,A( and ( )T,S satisfy the property E.A.

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Also every compatible pair is weakly compatible. Therefore a unique common fixed point follows from
Theorem B, under the restricted orbital completeness of the space X. It is interesting to note that the orbital
continuity is not needed to obtain a common fixed point.
Corollary(Theorem C, [5]): Let A, S and T be self-maps on X satisfying (2). Given Xx0 ∈ , suppose that
there is an orbit with choice (1) and the conditions (a) and (b) of Theorem A hold good at x0. If A is onto and the
condition (g) of Theorem B hold good, then A, S and T will have unique common fixed point.
Proof: Since A is onto we see that ( ) XXA = . In view of Remark 2.1, a unique common fixed point follows
from Theorem B.
References
[1] M. Aamri and EI D. I. Mountawaki, Some new common fixed point theorems under strict contractive
conditions, J. Math. Aanal. and Appl., 270 (2002), 181-188.
[2] Gerald Jungck, Compatible mappings and common fixed points, Int. Jour. Math. & Math. Sci. 9 (1986),
771-779.
[3] Gerald Jungck and B. E. Rhoades, Fixed point for set-valued functions with out continuity, Indian J. pure
appl. Math. 29 (3) (1998), 227-238.
[4] T. Phaneendra, Orbitalcontinuity and common fixed point, Buletini Shkencor, 3(4) (2011),375-380.
[5] Swatmaram and Phaneendra T, Common fixed points of Compatible self-mape using Asymptotic Regularity,
Acta Ciencia Indica, 33 (3) (2007), 1243-1247.

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